The following connections hold between \(H^1(p,w)\)  and \(E^2(p,w)\)
\(f\in E^2(p,w)\Rightarrow \int _c^{\cdot } f\in H^1(p,w) \quad \text{ and }\quad h\in H^1(p,w)\Rightarrow h^{\prime } \in E^2(p,w).\) 
Indeed, if \(f\in E^2(p,w)\) , we have \(f\in L^1_{\text{loc}}(]a,b[)\)  and so \(\int _c^{\cdot } f\in H^1(p,w)\)  by Lemma VIII.2 in [1]}. If \(h\in H^1(p,w)\) , there exists a continuous version of \(h\)  such that \(\int _c^{\cdot } h^{\prime }=h-h(c)\in L^2(p)\)  by Theorem VIII.2 in [1]}.
We deduce that the operators \(I:E^2(p,w)\rightarrow H^1_c(p,w)\)  and \(D:H^1_c(p,w)\rightarrow E^2(p,w)\)  are well defined. The statement \(I=D^{-1}\)  is obvious. The continuity of \(I\)  and \(D\)  follows from the choice of norms on \(E^2(p,w)\)  and \(H^1_c(p,w)\) .
In order to show that \(L^2(pw)\cap L^1_{\text{loc}}(]a,b[)= E^2(p,w)\) , we just have to show \(L^2(pw)\cap L^1_{\text{loc}}(]a,b[)\subset E^2(p,w)\)  as the other inclusion is evident. By Lemma VIII.2 in [1]}, every function in \(L^1_{\text{loc}}(]a,b[)\)  can be written as the weak derivative of a function \(h\in L^1_{\text{loc}}(]a,b[)\) . Assume by contradiction that there exists a function \(h\)  weakly differentiable such that \(\mathbb {E}_p[h]=0\) , \(\left\Vert h^{\prime }\right\Vert _{L^2(pw)}=1\)  but \(\left\Vert h\right\Vert _{L^2(p)}=\infty \) . For each \(n\in \mathbb {N}\) , set \(h_n(x)=\max \left\lbrace -n,\min \left\lbrace h,n\right\rbrace \right\rbrace \) .
We have \(\left\Vert h_n\right\Vert _{L^2(p)}\le n\)  and \(\left\Vert h_n^{\prime }\right\Vert _{L^2(pw)}\le \left\Vert h^{\prime }\right\Vert _{L^2(pw)}= 1\) , so that \(h_n\in H^1(p,w)\) . As \((h_n^2)\)  is an increasing sequence of functions and \(h_n\rightarrow h\)  a.e. we have by monotone convergence that \(\left\Vert h_n\right\Vert _{L^2(p)}\rightarrow \left\Vert h\right\Vert _{L^2(p)}=\infty \) . This is a contradiction with the fact that \(C(p,w)<\infty \) .
Finally, we deal with the four equivalences. The two first statements are equivalent because \(E^2(p,w)\)  and
\(H^1_c(p,w)\)  are homeomorphic by Proposition . The third
assertion entails the fourth one since \(L^2(pw)\cap L^1_{\text{loc}}(]a,b[)= E^2(p,w)\) .
The fourth one implies
the first one because \(L^2(pw)\)  is a Hilbert space. Indeed, it is
the case as soon as \(pw\)  is the density of a \(\sigma \) -finite measure
(see section 3.2 in [4]}), which is true since
\(pw\in L^1_{\text{loc}}(]a,b[)\) . It remains to show that the first
one entails the third one. By contradiction, assume that there
exists \(f\in L^2(pw)\setminus L^1_{\text{loc}}(]a,b[)\) . Define \(f_n(x)=\max \left\lbrace -n,\min \left\lbrace f,n\right\rbrace \right\rbrace \) .
We have \((f_n)\subset L^2(pw)\cap L^1_{\text{loc}}(]a,b[)=E^2(p,w)\)  and \(f_n\rightarrow f\)  in \(L^2(pw)\)  by dominated convergence. Hence, \(E^2(p,w)\)  isn't closed in \(L^2(pw)\) . Since \(E^2(p,w)\)  is a subspace of \(L^2(pw)\)  endowed with the same norm, this implies that it is not a Hilbert space.
